# pr.probabilidad – Conditioning in an irrelevant variable in a martingale control problem

Suppose I have two independent Brownian movements. $$B ^ 1_t, B ^ 2_t$$ Y $$mathbb F_t$$ Be the natural filtration generated by them. Leave $$T> 0$$ be a fixed finite number Leave $$q_t$$ be a $$[-1,1]$$ valued $$mathbb {F} _t$$ Martingale that controls the analyst.

Leave $$mathcal Q$$ be the set of $$[-1,1]$$ valued $$mathbb F_t$$ Martingales The control problem is: $$sup_ {q in mathcal Q} E_ {0, q_0} f (q_T, B ^ 1_T)$$

where $$q_0$$ is the value of $$q$$ at the time $$0$$. $$f ( cdot, cdot)$$ It is linear in each argument. As we can see, the objective function does not depend on $$B ^ 2_t$$.

I would guess that, given the linearity of $$f ( cdot, cdot)$$ in each of its arguments and without dependence on $$B ^ 2_t$$, that we can restrict attention to $$[-1,1]$$ valued $$sigma (B ^ 1_t)$$ where the martingales $$sigma (B ^ 1_t)$$ It is a coarser filtering generated only by $$B ^ 1_t$$.

Does that sound like a reasonable guess? And how can I argue this if it's true?